Post on 31-Dec-2015
Section 1.4
Continuity and One-sided Limits
Continuity – a function, f(x), is continuous at x = c only if the following 3 conditions are met:
1. is defined
2. exists
3.
Continuous on an open interval (a, b)A function is continuous on (a, b) only if it is continuous at every point in (a, b).
f c
limx cf x
limx cf x f c
Condition 1 is not met: hole in graph
c
f(c) is not defined
Condition 2 is not met: jump or asymptote
c
does not exist
c
does not exist limx cf x
lim
x cf x
Condition 3 is not met: hole in graph and function defined elsewhere.
c
L
f(c)
limx c
f c f x
Two Types of discontinuities
1. removable – function that can be made continuous at a point by redefining f(c).
ex: hole in graph
2. nonremovable – cannot redefine f(c) to make the function continuous.
ex: asymptote or jump in graph
Examples: Discuss the continuity of each.
1.
2
1f x
x
nonrem. discont. @ x = 0 (asymptote)
Examples: Discuss the continuity of each.
2.
2 2 1
1
x xf x
x
rem. disc. @ x = 1 (redefine f(1)=0)
1
Examples: Discuss the continuity of each.
3.
3
4, 2
, 2
x if xf x
x if x
nonrem. disc. @ x = 2 (jump)
2
6
8
Examples: Discuss the continuity of each.
4.
cosf x x
cont. on (-∞, ∞)
One-sided Limits
limit from the right
limit from the left
limx c
f x L
limx c
f x L
Examples: Evaluate each limit.
5. 2
2lim 4x
x
= 0
-2
Examples: Evaluate each limit.
6. 2
2lim 4x
x
= DNE
-2
Examples: Evaluate each limit.
7. 1
limx
x = 0
1
Examples: Evaluate each limit.
8. 2
limx
x = -2
-2
Existence of a LimitWhen , then lim lim
x c x cf x f x L
lim
x cf x L
Continuity on a closed interval [a, b]
f(X) is continuous on (a, b) and
lim limx a x b
f x f a and f x f b